If in particular we look at \(g=2\) , we see that the current lower bound is linear while the current best constructions for \(a=1\)  and \(h\)  divisible by 4 is \(n^{h/2}\)  [1]}. This shows that there is a large scope of improvement in either direction.
Any progress in this question is very interesting because as mentioned earlier the regime of small number of groups is important in practice for distributed storage [2]}.
